# What are the FULL classical electrodynamic equations?

samalkhaiat
Science Advisor
pellman said:
what I need then, I guess, is the Maxwell equations (got 'em) + the continuity equation (got it) + plus the continuous version of the Lorentz force in terms of $$\rho$$ and $$J$$ and depending on E and B (need it).

If someone could provide that last piece, I'd appreciate it.

You can write
$$m= \int \mu d^3x$$
$$e= \int \rho d^3x$$

in the Lorentz force

$$\frac{dP_{i}}{dt} = \int d^3x \mu \frac{dv_{i}}{dt}= \int d^3x [ \rho E_{i} + \rho \epsilon_{ijk} v_{j} H_{k}]$$

or

$$\mu \frac{dv_{i}}{dt}= \rho E_{i} +\epsilon_{ijk} J_{j} H_{k}$$

In 4-vector, you have

$$\mu \frac{dV^{\mu}}{ds}= \rho F^{\mu\nu} V_{\nu}$$

or

$$\mu \frac{dV^{\mu}}{dt}= F^{\mu\nu} J_{\nu}$$

This can be turned to conservation statement.

As for the Lagrangian, it is

$$\mathcal{L}= - \mu (1-v^2)^{1/2} -A_{\mu} J^{\mu} -\frac{1}{16\pi} F_{\mu\nu}F^{\mu\nu}$$

regards

sam

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Thank you very much, Sam. I hope you will be available for further questions in a couple of days after I have digested this much. I have decided to read Landau & Lifchitz Classical Theory of Fields (as someone mentioned above) and hopefully answer the question of this thread while doing so. I hadn't seen that series of books in ten years and had forgotten how good they are.

But in the meantime, what do we do with a Lagrangian like $$\mathcal{L}= - \mu (1-v^2)^{1/2} -A_{\mu} J^{\mu} -\frac{1}{16\pi} F_{\mu\nu}F^{\mu\nu}$$? If we treat each of the $$J^{\mu}$$ as independent, we wind up with

$$\frac{\partial\mathcal{L}}{\partial J^{\alpha}}-\partial_{\mu}\frac{\partial\mathcal{L}}{\partial( \partial_{\mu}J^{\alpha})}=0 \rightarrow A_{\alpha}=0$$

since the the derivatives of J do not appear in the Lagrangian. But that result doesn't make any sense. So we must have $$\frac{\partial J^{\alpha}}{\partial(\partial_{\mu}J^{\beta})}\neq 0$$ for some alpha, beta, mu. How does that work?

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pellman,

It is a great idea to read Landau & Lifchits, it contains an incredible broad range of physics with little-known but important and fundamental subjects all treated in a very rigourous and concise way.

You will not find the answer to your exact problem. However you will quickly remember how these varional principles are developped successively for mechanics, fields, and finally their coupling.

The last part, the coupling of fields and particles is -in fact- what you intend to translate in terms of charge and mass density instead of charge coordinates. Clearly, some tought is needed on how this could be done. But obviously, you need to identify the coordinates that make sense for this development. One simple possibility would be the coordinates of the partciles themselves ... and -naïvely- I could imagine that mapping those coordinates on the coordinates you want to use (densities) would do the job. Honestly, I am very curious to know more about that!

Michel

Poscriptum

Mapping the coordinates of a swarm of particles on the space of density functions does not look as a one-to-one relation: there are probably more possible density functions than possibles swarms of particles (note I am not a mathematician). Maybe this will not be a real problem by assuming some smoothing hypothesis (like going from atoms to fluids mathematically).

Note also that a fluid model might not always be possible. I am thinking for example to a magnetised plasma in a tokamak. At high temperatures, some ions can have cyclotron trajectories that circle largely (not always circles!) and cover regions of different plasma densities and temperature. In this case, it is hard to see how a fluid picture can fit with this physics.

Maybe what you are thinking of is more a classical approach on the border of quantum mechanics: a model where particles have a finite size. If this is your goal, then I think you need to bring additional physics which is basically unknown and that should somehow match with QM on the border of CM and QM. Certainly this is not part of the current classical physics, since this is answered by quantum mechanics. Consider for example that the width of the distribution associated to an electron might be a dynamic variable too, and classical physics has always been silent on that.

Reading L&L could be a source of inspiration. For each variational principle introduced in this book, they explain why the Lagrangian (action) should have the assumed structure. This may guide you to decide about a possible structure for your own model.

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pellman,

When taking the functional derivative of your Lagragian

$$\mathcal{L}= - \mu (1-v^2)^{1/2} -A_{\mu} J^{\mu} -\frac{1}{16\pi} F_{\mu\nu}F^{\mu\nu}$$​

have you tried what happens if you assume the mass and charge $$\mu$$ densities and $$J^{0}$$ are related?
The e/m ratio should provide the relation, unusual in a "fluid theory".

Note also the problem with the speed $$v^{\mu}$$ or $$v^{ }$$. It should also be related to the currents $$J^{\mu}$$ to close the variational principle. Isn't it?

This may go in the right direction, but this may also rise more questions.

Michel

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Thanks, Michel. I'm going to let this question go for now and keep a lookout for it in my general reading. The surprising thing is that it doesn't have a ready, short answer.

Thanks for the help.

Todd